Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519083 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1747 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,18)(12,22)(13,16)(14,17)(15,20)(19,24)(21,23);(1,3,5,2)(4,6,8,7)(9,10,13,22,14,17,12,16)(11,15,21,24,19,23,20,18);(1,4,5,8)(2,6,3,7)(11,19)(15,23)(18,24)(20,21);(2,3)(6,7)(9,10)(11,15)(12,16)(13,22)(14,17)(18,20)(19,23)(21,24);(9,11,13,21,14,19,12,20)(10,15,22,24,17,23,16,18)Polycyclic presentation:
2^9|a^2=g,b^2=fgi,c^2=g,e^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7453139);